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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> is <a href="glib_002.html#V1" target="_self">connected</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1 );<br/>
<a NAME="E1:77"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">G</font> being <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>  st <font color="Olive">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">G</font>]
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:77_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">Gk</font> being <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>  st <font color="Olive">Gk</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Gk</font>]
 ;<br/><b>let </b><font color="Maroon">Gk1</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>assume </b><a NAME="E2:77_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">Gk1</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">Gk1</font> is <a href="glib_002.html#V1" target="_self">connected</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_1_1"/>
<font color="Maroon">Gk1</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:77_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:77_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Gk1'</font> = <font color="Maroon">Gk1</font> as  <a href="glib_000.html#V2">finite</a> non <a href="glib_000.html#V4">trivial</a> <a href="glib_002.html#V1" target="_self">connected</a> <a href="glib_000.html#NM2">_Graph</a> <b>by </b><i>, <a class="ref" href="glib_000.html#T29">GLIB_000:29</a></i>;<br/><b>consider </b><font color="Maroon">v1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gk1'</font>, <font color="Maroon">v2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gk1'</font><b> such that </b><br/><a NAME="E5:77_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font> &amp; not <font color="Maroon">v1</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> &amp; not <font color="Maroon">v2</font> is <a href="glib_002.html#V5" target="_self">cut-vertex</a> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">Gkb</font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">Gk1'</font>,<font color="Maroon">v1</font>;<br/><a NAME="E7:77_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">Gkb</font> is <a href="glib_002.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="ref" href="glib_002.html#D2" target="_self">Def2</a>, </i>;<br/><a NAME="E8:77_1_1"/><i><font color="Green">E62</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">Gkb</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">Gkb</font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Gk1</font> <a href="glib_000.html#K18">.size()</a>  )
 <b>by </b><i>, <a class="ref" href="glib_000.html#T51">GLIB_000:51</a></i>;<br/><a NAME="E9:77_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">Gkb</font> <a href="glib_000.html#K18">.size()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Gk1</font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E10:77_1_1"/>
<font color="Maroon">Gkb</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E11:77_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Gk1</font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="glib_002.html#D3" target="_self">Def3</a>, </i>;<br/><a NAME="E12:77_1_1"/>
not <font color="Maroon">v1</font> is <a href="glib_000.html#V10">isolated</a>
 <b>by </b><i/>;<br/><a NAME="E13:77_1_1"/><b>then </b>
<font color="Maroon">v1</font> <a href="glib_000.html#K29">.edgesInOut()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#D51">GLIB_000:def 51</a></i>;<br/><a NAME="E14:77_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/><a NAME="E15:77_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E16:77_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E17:77_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Gk1</font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="glib_000.html#K29">.edgesInOut()</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E18:77_1_1"/>
<font color="Maroon">Gk1</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">Gk1</font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:77_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">Gk1</font>]
 ;<br/></div><b>end;</b></div>
<a NAME="E3:77"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">G</font> being <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">G</font>]
 
<b>from </b><i><a class="ref" href="glib_000.html#S1">GLIB_000:sch 1</a>(, <a class="ref" href="glib_002.html#D1" target="_self">Def1</a>);<br/></i>
<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_002.html#V1" target="_self">connected</a> <a href="glib_000.html#NM2">_Graph</a>;<br/>

<b>thus </b><a NAME="E4:77"/>
<font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/>


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